Nonlinear Valley Nernst Effect (NVNE)

• Nonlinear Valley Nernst Effect (NVNE) is a second-order thermoelectric response that generates valley currents in 2D multivalley systems via quadratic temperature gradients.
• It leverages intrinsic band geometric quantities, such as the quantum metric and Berry connection polarizability dipole, to produce valley-odd transverse signals even when linear responses vanish.
• Experimental manifestations include transverse thermovoltages and pure valley currents in materials like strained bilayer graphene and PT-symmetric Dirac systems, offering tunable, relaxation-time–independent effects.

The nonlinear valley Nernst effect (NVNE) is a transverse, second-order thermoelectric response in multivalley crystals, especially two-dimensional systems with inequivalent $K$ and $K'$ valleys, in which a temperature gradient generates a valley current or, equivalently in some formulations, a valley-contrasting transverse magnetization current that scales quadratically with the thermal drive. Current work distinguishes an intrinsic NVNE, independent of the relaxation time, from Berry-curvature-driven and Drude-like nonlinear thermoelectric terms. In the 2025 literature, the intrinsic effect is formulated either through the quantum metric and temperature-gradient-induced orbital magnetization, or through the Berry connection polarizability dipole; in both cases it survives in settings where linear valley Nernst or valley Hall responses are symmetry-forbidden (Sharma et al., 27 Jan 2025, Wu et al., 2 Sep 2025, Zhang et al., 27 Aug 2025).

1. Definition and response structure

A standard valley-current definition is

$$\mathbf{J}^v \equiv \mathbf{J}^{\mathrm K}-\mathbf{J}^{\mathrm K'},$$

and the second-order valley thermoelectric response can be written as

$$j_i^{v}=\alpha^{v(2)}_{ijk}\,(\nabla_jT)(\nabla_kT),$$

or, in the thermal-field notation $E^T=-\nabla T/T$,

$$j_\beta^{(2)}=\alpha_{\beta;\gamma\delta}^T\,E^T_\gamma E^T_\delta.$$

The defining feature is that the transverse response is valley-odd: the valley-resolved component changes sign between $\tau=\pm1$, corresponding to $K$ and $K'$ (Wu et al., 2 Sep 2025, Sharma et al., 27 Jan 2025).

NVNE is distinct from the linear valley Hall effect (LVHE) and linear valley Nernst effect (LVNE). LVHE is a first-order transverse valley current driven by an electric field and controlled by Berry curvature $\Omega_n(\mathbf{k})$. LVNE is the thermal analogue, likewise first order and Berry-curvature-based. In PT-symmetric systems, $K'$0 identically, so those linear responses vanish, whereas NVNE can remain finite because its intrinsic part is built from PT-even band-geometric quantities (Sharma et al., 27 Jan 2025).

Response	Driving field	Geometric control
LVHE	Electric field	Berry curvature $K'$1
LVNE	$K'$2	Berry curvature $K'$3 or orbital angular momentum
NVNE	$K'$4 or $K'$5	Quantum metric, thermoelectric orbital magnetization, or Berry connection polarizability dipole

In strained bilayer graphene, the response is explicitly Nernst-type: with strain along $K'$6, the surviving element is

$K'$7

so a temperature gradient along $K'$8 induces a transverse valley current along $K'$9 (Wu et al., 2 Sep 2025).

2. Intrinsic band geometry and microscopic mechanisms

One 2025 formulation identifies the intrinsic NVNE with the quantum metric $$\mathbf{J}^v \equiv \mathbf{J}^{\mathrm K}-\mathbf{J}^{\mathrm K'},$$0, the symmetric part of the quantum geometric tensor. For band $$\mathbf{J}^v \equiv \mathbf{J}^{\mathrm K}-\mathbf{J}^{\mathrm K'},$$1,

$$\mathbf{J}^v \equiv \mathbf{J}^{\mathrm K}-\mathbf{J}^{\mathrm K'},$$2

with $$\mathbf{J}^v \equiv \mathbf{J}^{\mathrm K}-\mathbf{J}^{\mathrm K'},$$3. In the quantum-kinetic treatment, the second-order current takes the form

$$\mathbf{J}^v \equiv \mathbf{J}^{\mathrm K}-\mathbf{J}^{\mathrm K'},$$4

and solving the Liouville equation in the relaxation-time approximation yields an intrinsic, $$\mathbf{J}^v \equiv \mathbf{J}^{\mathrm K}-\mathbf{J}^{\mathrm K'},$$5-independent conductivity tensor $$\mathbf{J}^v \equiv \mathbf{J}^{\mathrm K}-\mathbf{J}^{\mathrm K'},$$6 expressible through band-resolved quantum-metric objects $$\mathbf{J}^v \equiv \mathbf{J}^{\mathrm K}-\mathbf{J}^{\mathrm K'},$$7 and $$\mathbf{J}^v \equiv \mathbf{J}^{\mathrm K}-\mathbf{J}^{\mathrm K'},$$8 (Sharma et al., 27 Jan 2025).

A closely related semiclassical formulation writes the intrinsic nonlinear thermoelectric tensor as

$$\mathbf{J}^v \equiv \mathbf{J}^{\mathrm K}-\mathbf{J}^{\mathrm K'},$$9

with

$$j_i^{v}=\alpha^{v(2)}_{ijk}\,(\nabla_jT)(\nabla_kT),$$0

and

$$j_i^{v}=\alpha^{v(2)}_{ijk}\,(\nabla_jT)(\nabla_kT),$$1

This makes the quantum-metric origin explicit and separates the intrinsic contribution from Berry-curvature and Drude terms, which scale as $$j_i^{v}=\alpha^{v(2)}_{ijk}\,(\nabla_jT)(\nabla_kT),$$2 and $$j_i^{v}=\alpha^{v(2)}_{ijk}\,(\nabla_jT)(\nabla_kT),$$3, respectively (Wu et al., 2 Sep 2025).

A second 2025 formulation expresses intrinsic NVNE through the Berry connection polarizability dipole (BCPD). The gauge-invariant tensor

$$j_i^{v}=\alpha^{v(2)}_{ijk}\,(\nabla_jT)(\nabla_kT),$$4

enters the intrinsic valley tensor as

$$j_i^{v}=\alpha^{v(2)}_{ijk}\,(\nabla_jT)(\nabla_kT),$$5

This formulation emphasizes that the intrinsic response is a Fermi-surface property of band quantum geometry and is independent of scattering (Zhang et al., 27 Aug 2025).

The thermoelectric orbital-magnetization channel supplies the valley discriminator in PT-symmetric systems. The temperature-gradient-induced orbital magnetic moment $$j_i^{v}=\alpha^{v(2)}_{ijk}\,(\nabla_jT)(\nabla_kT),$$6 in strained bilayer graphene and the orbital magnetic moment $$j_i^{v}=\alpha^{v(2)}_{ijk}\,(\nabla_jT)(\nabla_kT),$$7 in the PT-symmetric quantum-kinetic treatment both acquire opposite signs in the two valleys. In that sense, the thermoelectric correction to orbital magnetization plays a role analogous to orbital angular momentum in the linear valley Hall picture (Wu et al., 2 Sep 2025, Sharma et al., 27 Jan 2025).

At low temperature, nonlinear Mott-type relations connect thermal and electric second-order tensors. One relation is

$$j_i^{v}=\alpha^{v(2)}_{ijk}\,(\nabla_jT)(\nabla_kT),$$8

while another is

$$j_i^{v}=\alpha^{v(2)}_{ijk}\,(\nabla_jT)(\nabla_kT),$$9

These results show that the intrinsic thermal response is systematically linked to nonlinear valley Hall transport, although the two papers emphasize different geometric kernels (Wu et al., 2 Sep 2025, Zhang et al., 27 Aug 2025).

3. Symmetry conditions and tensor selection rules

A central symmetry result is that NVNE is allowed under conditions that suppress linear valley responses. In PT-symmetric systems,

$E^T=-\nabla T/T$0

so PT enforces $E^T=-\nabla T/T$1, killing LVHE and LVNE. By contrast, the quantum metric satisfies

$E^T=-\nabla T/T$2

hence remains PT-even and can support a finite intrinsic second-order response (Sharma et al., 27 Jan 2025).

In globally inversion- and time-reversal-symmetric systems, the Berry-curvature contribution vanishes pointwise,

$E^T=-\nabla T/T$3

but the intrinsic valley difference can remain finite because time reversal maps $E^T=-\nabla T/T$4 to $E^T=-\nabla T/T$5 without forcing the single-valley tensor to vanish. Thus $E^T=-\nabla T/T$6 and $E^T=-\nabla T/T$7 may be equal in magnitude and opposite in sign, so the total charge current cancels while a pure valley current survives (Wu et al., 2 Sep 2025).

Local rather than global symmetry controls the allowed tensor components. In two dimensions, the largest local symmetry compatible with a nonzero intrinsic NVNE is a single mirror. With strain along $E^T=-\nabla T/T$8 in bilayer graphene, the surviving local symmetry is $E^T=-\nabla T/T$9, and the only allowed intrinsic valley component is

$$j_\beta^{(2)}=\alpha_{\beta;\gamma\delta}^T\,E^T_\gamma E^T_\delta.$$0

Additional rotational symmetries such as $$j_\beta^{(2)}=\alpha_{\beta;\gamma\delta}^T\,E^T_\gamma E^T_\delta.$$1 or $$j_\beta^{(2)}=\alpha_{\beta;\gamma\delta}^T\,E^T_\gamma E^T_\delta.$$2 with $$j_\beta^{(2)}=\alpha_{\beta;\gamma\delta}^T\,E^T_\gamma E^T_\delta.$$3 forbid the response (Wu et al., 2 Sep 2025).

A complementary selection-rule formulation uses the pseudotensor

$$j_\beta^{(2)}=\alpha_{\beta;\gamma\delta}^T\,E^T_\gamma E^T_\delta.$$4

which transforms under a point-group operation $$j_\beta^{(2)}=\alpha_{\beta;\gamma\delta}^T\,E^T_\gamma E^T_\delta.$$5 as

$$j_\beta^{(2)}=\alpha_{\beta;\gamma\delta}^T\,E^T_\gamma E^T_\delta.$$6

with $$j_\beta^{(2)}=\alpha_{\beta;\gamma\delta}^T\,E^T_\gamma E^T_\delta.$$7 for valley-preserving and $$j_\beta^{(2)}=\alpha_{\beta;\gamma\delta}^T\,E^T_\gamma E^T_\delta.$$8 for valley-switching operations. In point group $$j_\beta^{(2)}=\alpha_{\beta;\gamma\delta}^T\,E^T_\gamma E^T_\delta.$$9, this allows $\tau=\pm1$0, whereas three- and fourfold rotational symmetries typically forbid the same components (Zhang et al., 27 Aug 2025).

Anisotropy or tilt is therefore not a minor perturbation but a symmetry-enabling ingredient. In the tilted Dirac model of the PT-symmetric study, $\tau=\pm1$1 kills NVNE. In strained bilayer graphene, the temperature gradient must be applied perpendicular to the strain direction. These results establish NVNE as a symmetry-selected anisotropic transport coefficient rather than a generic consequence of valley degeneracy (Sharma et al., 27 Jan 2025, Wu et al., 2 Sep 2025).

4. Canonical models and material realizations

The prototypical PT-symmetric model is an anisotropic tilted Dirac semimetal with

$\tau=\pm1$2

Its Berry curvature vanishes under PT symmetry, but the band-resolved quantum metric remains finite. For the valence band,

$\tau=\pm1$3

The intrinsic NVNE tensor component is obtained analytically as

$\tau=\pm1$4

with $\tau=\pm1$5 for conduction and valence bands. In the gapless limit,

$\tau=\pm1$6

independent of $\tau=\pm1$7. The valley-dependent thermoelectric orbital magnetic moment is

$\tau=\pm1$8

and the integrated valley orbital magnetization is

$\tau=\pm1$9

These expressions make the valley sign, tilt dependence, and band dependence explicit (Sharma et al., 27 Jan 2025).

The same paper extends the analysis to PT-symmetric bilayer graphene with trigonal warping and to the organic conductor $K$0. In bilayer graphene, integrating the thermoelectric orbital magnetization yields

$K$1

while in $K$2,

$K$3

In both materials the sign reverses between valleys, and in the organic compound the sign of the NVNE also reverses between valence and conduction bands (Sharma et al., 27 Jan 2025).

A distinct but related realization is uniaxially strained AB-stacked bilayer graphene, described by

$K$4

with

$K$5

For strain along $K$6, only $K$7 survives, so $K$8 is required to generate $K$9. The intrinsic coefficients satisfy

$K'$0

leaving zero charge current and a pure valley current. Changing from compressive ($K'$1) to tensile ($K'$2) strain reverses the sign of $K'$3, with peaks near Lifshitz transitions around $K'$4 and $K'$5 (Wu et al., 2 Sep 2025).

First-principles work on bilayer $K'$6 provides an inversion-asymmetric realization. With point group $K'$7, two well-separated valleys on the $K'$8 axis, and $K'$9 allowed by symmetry, the intrinsic NVNE component $\Omega_n(\mathbf{k})$0 peaks near $\Omega_n(\mathbf{k})$1 with magnitude on the order of $\Omega_n(\mathbf{k})$2 at $\Omega_n(\mathbf{k})$3 (Zhang et al., 27 Aug 2025).

5. Scaling laws, tunability, and experimental signatures

Parameter dependence is unusually transparent in the tilted Dirac and strained bilayer graphene models. In the PT-symmetric tilted Dirac case, the response grows with the tilt or anisotropy parameter $\Omega_n(\mathbf{k})$4, vanishes for $\Omega_n(\mathbf{k})$5, decreases as the gap $\Omega_n(\mathbf{k})$6 increases, and changes sign between conduction and valence bands through the factor $\Omega_n(\mathbf{k})$7. For finite $\Omega_n(\mathbf{k})$8, the analytical expression depends on $\Omega_n(\mathbf{k})$9 and $K'$00; in the gapless limit it becomes independent of $K'$01 (Sharma et al., 27 Jan 2025).

Temperature scaling requires care because different papers normalize the response differently. In the $K'$02 convention, the intrinsic coefficient in the PT-symmetric Dirac model scales as $K'$03. In the $K'$04 convention used for strained bilayer graphene and for the BCPD theory, the low-temperature intrinsic coefficients obey Sommerfeld-type relations and scale as $K'$05. This difference reflects tensor conventions rather than a contradiction in the existence of the effect (Sharma et al., 27 Jan 2025, Wu et al., 2 Sep 2025, Zhang et al., 27 Aug 2025).

Several experimental observables have been proposed. Direct transverse thermovoltage measurements apply $K'$06 longitudinally and read out a transverse signal; a nonzero response in globally PT-symmetric samples, where linear Nernst transport is absent, is a primary signature. Nonlocal valley transport uses valley filters or injectors and tests sign reversal under valley polarization switching. Scanning SQUID or NV-center magnetometry can image edge magnetization currents, exploiting $K'$07. Kerr rotation, circular dichroism, and nonreciprocal directional dichroism have been proposed as valley-sensitive optical probes of edge accumulation or magnetic-quadrupole structure (Sharma et al., 27 Jan 2025, Wu et al., 2 Sep 2025, Zhang et al., 27 Aug 2025).

The nonlocal theory for intrinsic NVNE predicts a second-harmonic signal with characteristic $K'$08 scaling,

$K'$09

and a ratio

$K'$10

where $K'$11 is the thermopower and $K'$12 is the Lorenz number. In bilayer $K'$13, using $K'$14, $K'$15, $K'$16, and $K'$17, the estimated second-harmonic nonlocal voltage is

$K'$18

and at $K'$19 the ratio $K'$20 is $K'$21, so the signal is dominated by the direct NVNE pathway (Zhang et al., 27 Aug 2025).

The PT-symmetric study gives order-of-magnitude estimates in a different language. With $K'$22 of order $K'$23–$K'$24 and $K'$25–$K'$26, the gapless NVNE coefficient can generate measurable transverse voltages under strong thermal gradients. For $K'$27–$K'$28 and device widths of $K'$29–$K'$30, the resulting magnetization current density is stated to correspond to microampere-per-meter scale transverse currents and microvolt-to-millivolt thermovoltages, although exact SI conversion depends on band parameters and geometry (Sharma et al., 27 Jan 2025).

6. Relation to adjacent nonlinear Nernst effects and current limitations

A recurring misconception is that all nonlinear Nernst effects are equivalent. Earlier work on strained transition-metal dichalcogenides analyzed a nonlinear anomalous Nernst current in time-reversal-symmetric but inversion-broken systems, controlled by Berry curvature near the Fermi surface and by the pseudovector

$K'$31

There the valley mechanism is explicit, but the two valley contributions add to a net transverse charge current rather than a pure valley current (Yu et al., 2019).

Likewise, the 2019 study of bilayer $K'$32 treated the nonlinear anomalous Nernst effect in an inversion-broken, time-reversal-symmetric material and found a Berry-curvature-controlled second-order charge current proportional to a Fermi-surface quantity $K'$33. Valley-resolved contributions can be defined, but the reported observable is the net charge response, not a symmetry-protected pure valley current (Zeng et al., 2019).

The 2020 quantum-kinetic treatment introduced a thermoelectric Berry-curvature dipole $K'$34 and showed that, in inversion-broken crystals, second-order thermoelectric transport may contain two intrinsic Berry-curvature-linked terms. That analysis also stated that charge and pure valley responses depend on whether the relevant anisotropy or tilt is valley-odd or valley-even: valley-odd tilt tends to make charge currents add, whereas valley-even deformations can leave $K'$35 but $K'$36 (Li, 2020). This suggests that the modern NVNE literature sits at the intersection of two threads: Berry-curvature-based nonlinear anomalous thermoelectric transport in inversion-broken systems, and quantum-metric-based or BCPD-based pure valley transport that survives global inversion and time-reversal symmetry.

The present theories also share limitations. The PT-symmetric quantum-kinetic and semiclassical treatments focus on the intrinsic, clean-limit contribution, neglecting disorder-dependent skew-scattering and side-jump corrections in the main result. Strong intervalley scattering can reduce the net NVNE by relaxing valley imbalance. The strained bilayer-graphene analysis assumes weak, slowly varying temperature gradients, a low-energy $K'$37 regime, and no intervalley scattering. The first-principles nonlocal theory similarly assumes long intervalley diffusion length and uses transport-scaling arguments to separate intrinsic and extrinsic terms (Sharma et al., 27 Jan 2025, Wu et al., 2 Sep 2025, Zhang et al., 27 Aug 2025).

Taken together, these works define NVNE as a genuine nonlinear valley caloritronic effect rather than a simple extension of linear Nernst physics. Its most distinctive forms are the intrinsic, relaxation-time-independent responses permitted by band quantum geometry, selected by local mirror symmetry, amplified by anisotropy or trigonal warping, and observable even when Berry curvature and all linear valley Hall or Nernst signals vanish globally.